I struggle to imagine you can get a toilet in there. Looking at a washer and dryer, you need to have them along the long wall. The average washing is 30" deep, possible more. That's 2.5 feet, leaving you with 3.5 feet clearance. Unless you have a top loading washer, when the door is open, it will be probably 56" or over 4.5 feet, leaving you with a foot and a half of clearance.

Now, have some pipes and other stuff along that wall, so you may not be able to put your appliances right up against the wall, meaning you may have less clearance. So, a washer and dryer are going to be tight.

Also, is the 13' length the length of the room, which has a lot of stuff taking up the back of it or is it the open space up to the heating system?

What's pretty clear to me is that if you want both a bathroom and a laundry room, you probably aren't going to be able to put a wall between them.

I see what you mean about the clearance. A 6' width really doesn’t leave much room once you account for the depth of the washer/dryer and the space needed to move around them. Even a compact powder room usually needs enough width for a toilet and some minimal clearance in front, which might be tough if appliances are along the same run.

Maybe the only workable option would be stacking the washer/dryer or using compact units, then placing a very small powder room at one end of the space. But with mechanical equipment and piping already there, it would probably depend a lot on the exact layout.